The problem is therefore what happens when one descends in regularity of \(V^i\)  further beyond Lipschitz continuity. In fact, there are many cases when Cauchy problems (REF ) admit unique solutions for every or just for a.e. intial datum.
For instance, as shown in [1]}, for a vector field \(V\in W^{1,p}(\mathbb {R}^d;\mathbb {R}^d)\)  with \(p>d\)  and \(\operatorname{div}V\)  bounded, a unique flow \(x\mapsto F_t(x)\)  is defined for a.e. \(x\in \mathbb {R}^d\)  and all \(t\in \mathbb {R}\)  as \(F_t(x):= y(t)\)  where \(y(\cdot )\)  is the unique solution to the ODE
\(\dot{y}=V(y)\) 
